10-1.1 Lesson Presentation

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Chapter 10Estimating With ConfidenceMs. PlataAP Statistics



Section 10.1

Confidence Intervals: The Basics



Statistical Inference

The goal of statistical inference is to infer from the sample datasome conclusion about the population.

We can’t be certain that our conclusions are correct - a differentsample might lead to different conclusions.

Statistical inference uses the language of probability to expressthe strength of our conclusions




The two most common types of formal statistical inference:

Confidence Intervals for estimating the value of a populationparameter.

Significance Tests assess the evidence for a claim about a

population (Chapter 11)

Both types of inference are based on the sampling distribution ofstatistics




Inference is most reliable when the data are produced by a properlyrandomized design.

When you use statistical inference, you are acting as if the data are arandom sample or come from a randomized experiment.



Recall from Chapter 9

The mean of the sampling distribution of x-bar is the same as theunknown mean μ of the entire population.

The standard deviation of x-bar is .

The Central Limit Theorem tells us that the mean x-bar of x > 30 hasa distribution that is close to Normal.



Confidence Intervals: The Basics

Read example 10.1 on p. 618 and p. 619

Read example 10.2 on p. 620

A confidence interval is a set (range) of population values for which

our found sample value is likely.

Read example 10.3 on p. 621



Confidence Intervals



• The center x-bar of eachinterval is marked by adot.

•The arrows on eitherside of the dot span theconfidence interval.

• The distance from thedot to the end of an arrowis the margin of error forthat interval.

• 24 of these 25 intervals(96%) cover the truevalue of μ. If we took all

possible samples, 95% ofthe resulting confidenceintervals would contain μ.



Assignment #1

10.1 (to be done in class)

10.2

10.5

10.6



Confidence Interval for Population

Mean (Whenσ

is known)

Be sure to check that these conditions for constructing aconfidence interval for μ are satisfied before you perform any

calculations



AP Tip

A common error students make on the AP Exam is to fail to identify

the conditions by which they are justified in constructing a confidenceinterval.

This is understandable, as many questions have simply directedstudents to “construct a confidence interval” and there are no specific

directions to first justify it.

However, students MUST show that the conditions exist to construct avalid confidence interval in order to get full credit on a question.



Finding Critical Values z*

Read example 10.4 on how to find z*



Critical Values z*

Confidence

Level Tail Area z*

90% 0.05 1.645

95% 0.025 1.960

99% 0.005 2.576

The most common z*:

Values z* that mark off a specified area under the standardNormal curve are often called critical values of the distribution.



Critical Values



Confidence Interval

Used to estimate the unknown population parameter.



Confidence Interval



Example 10.5

Constructing a confidence interval for μ

p. 630

This procedure will be used throughout the rest of the class whendoing inference.

All FOUR steps in the process must be present in an inference

problem on the AP exam in order to receive full credit.

In Step 4 of the procedure, note that we do NOT make a probabilitystatement about a found confidence interval and that the interpretationmust be in the context of the problem.



Inference Toolbox

Step 1: Parameter . Identify the population of interest and theparameter you want to draw conclusions about.

Step 2: Conditions . Choose the appropriate inference procedure.Verify conditions for using it.

Step 3: Calculations. If the conditions are met, carry out the inferenceprocedure.

confidence interval = estimate +- margin of error

Step 4: Interpretation. Interpret your results in the context of theproblem. Remember the “three C’s”: conclusion, connection, andcontext



Example

A test for the level of potassium in theblood is not perfectly precise. Suppose

that repeated measurements for the sameperson on different days vary Normally withσ = 0.2. A sample of three have a mean of

3.2. What is a 90% confidence interval forthe mean potassium level?



Example Continued

95% confidence interval?

✤99% confidence interval?

✤ What happens to the interval as the confidence level increases?

The interval gets wider as the confidence level increases



Interpretation (Memorize!!!)

We are ________% confident that the truemean context lies within the interval

______ and ______.

We are 95% confident that the true meanpotassium level in the blood lies within theinterval 2.97 and 3.43.



Assignment #2

10.7

10.9

10.11

10.12



Margin of Error Gets SmallerWhen:

z* gets smaller (smaller confidence level C).

σ gets smaller (less variation in the population).

n gets larger (to cut the margin of error in half, n must be 4 times asbig).

NOTE: The researcher cannot control σ. She/he can control theconfidence level and the sample size.



How Confidence IntervalsBehave

We want high confidence and a small margin of error.

A small margin of error says that we have pinned down the parameter

quite precisely.

The margin of error is (can be used to find sample size):

Only by manipulating n you can control the margin of error.

Always round UP to the nearest person!!!



Determining Sample Size



Example

The heights of ASD male students is Normally distributed with σ= 2.5 inches. How large a sample is necessary to be accuratewithin + .75 inches with 95% confidence?



Caution!!!!!!

The margin of error only covers random sampling error. It does NOTcover errors due to:

The data not being a SRS from the population.

Data from more complex sampling designs (stratified, etc.)

Bias in data (wording, nonresponse, etc.)

Watch for outliers or strong skewness.

Must know σ

p.636 - 637



What Confidence Does NOT Say:

We are 95% confident that the mean SAT Math score for all Texas high school students is between 452 and 470.

✴DOES NOT SAY the probability is 95% that the true mean fallsbetween 452 and 470.

✴DOES SAY the numbers were calculated by a method that gives

correct results in 95% of all possible samples.

We either captured the true mean in our interval, or we didn’t. (P=1 or P=0) The probability describes how often the METHOD gives correct answers.



Calculator Tip

To calculate Confidence Intervals in the Calculator, follow theTechnology Toolbox on p. 641 - 642



Assignment

10.13

10.15

10.16

10.17

10.18

10.22

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10-1.1 Lesson Presentation